Flexibility of the isometric immersion system in arbitrary dimension and codimension and the energy scaling of prestrained thin films

We prove that, for a given $C^{r,\beta}$-regular Riemann metric posed on a $d$-dimensional domain, every short immersion into the Euclidean space $\mathbb{R}^{d+k}$, can be uniformly approximated by exact isometric immersions of regularity $C^{1,\alpha}$ for any $\alpha<\alpha_0=\min\{\frac{r+\beta}{2}, \frac{1}{1+d(d+1)/k}\}$.

Our theorem recovers several previously known results as special cases.

The novelty thereof lies in providing a unified flexibility statement for arbitrary dimensions $d$ and codimensions $k$, while also treating the so far uncharted range $k\in (1, \frac{d(d+1)}{2}-d+1)\setminus \{d\}$, where no corresponding general result was previously available.

Our threshold flexibility exponent $\alpha_0$ agrees with that previously obtained for the closely related Monge-Ampère system.

As an application, we prove a new estimate in the quantitative immersability of thin prestrained films, setting the scaling exponent of the infimum of non-Euclidean energies in presence of an arbitrary prestrain metric, and in the limit of the film's vanishing thickness, at $\frac{4\alpha_0}{\alpha_0+1}$.